Abstract
A new method for the solution of population balance equations (PBE) describing the micro-processes such as nucleation, growth, aggregation of particle swarms in a multiphase system is proposed. The method is based on the fixed pivot moment and allows arbitrary number of moments to be tracked simultaneously. By expressing PBEs for both batch and continuous operations in a general form, and using weighted residual method to derive the moment equations, different moments can be tracked directly. The numerical density function is assumed to be a summation of several weighted Dirac Delta functions, and the integral and derivative terms in PBEs are transformed to a summation in order to reduce computational cost. Simulations of a batch nucleation-growth process and a continuous aggregation-growth process have demonstrated good agreement with the corresponding analytical solutions, with relative errors less than 10−8%. Simulation of a combined nucleation-growth-aggregation process, which does not have an analytical solution, is also included, so as to reproduce the micro-behaviors of such a complex system, demonstrating the feasibility and reliability of this method.
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Gu Z L, Su J W, Li Y, Feng S Y. Behaviors of the dispersed phase in the multiphase system and population balance model (in Chinese). Chem Reac Eng Tech, 2007, 23(2): 162–167
Su J W, Gu Z L, Li Y. Direct quadrature method of moment on characteristic volume for population balance equation (in Chinese). J Xi’an Jiaotong Uni, 2007, 41(5): 621–624
McGraw R. Description of aerosol dynamics by the quadrature method of moments. Aerosol Sci Technol, 1997, 27(2): 255–265
Fan R, Marchision D-L, Fox R-O. Application of the direct quadrature method of moments to polydisperse gas-solid fluidized beds. Powder Technol, 2004, 139(1): 7–20
Kumar S, Ramkrishna D. On the solution of population balance equations by discretization.i. fixed pivot technique. Chem Eng Sci, 1996, 51(8): 1311–1332
Golub G-H, Vanloan C-F. Matrix Computations. 3th ed. Baltimore: The Johns Hopkins Univ Press, 1996. 183–188
Marchisio D-L, Fox R-O. Solution of population balance equations using the direct quadrature method of moments. J Aerosol Sci, 2005, 36(1): 43–73
Su J W, Gu Z L, Li Y, Feng S Y. Study on quadrature method of moment of volume for population balance model for dispersed phase system (in Chinese). Chem Reac Eng Tech, 2007, 23(6): 518–524
Ramabhadran T-E, Peterson T-W, Seinfeld J-H. Dynamics of aerosol coagulation and condensation. AIChE J, 1976, 22(5): 840–851
Su J W, Gu Z L, Li Y, Feng S Y, Xu X-Y. Solution of population balance equation using quadrature method of moments with an adjustable factor. Chem Eng Sci, 2007, 62(21): 5897–5911
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Supported by the National Basic Research Program of China (Grant No. 2004CB720208), National Natural Science Foundation of China (Grant Nos. 40675011 and 10872159) and Key Laboratory of Mechanics on Disaster and Environment in Western China (Grant No. 200707)
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Gu, Z., Su, J., Jiao, J. et al. Simulation of micro-behaviors including nucleation, growth, and aggregation in particle system. Sci. China Ser. B-Chem. 52, 241–248 (2009). https://doi.org/10.1007/s11426-009-0014-2
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DOI: https://doi.org/10.1007/s11426-009-0014-2